express in sinx
1 1
---------- + --------
cscx + cotx cscx - cotx
and express in cosx
1 + cot x
------- - sin^2x
cscx = 1/sinx so what do i do w. that extra one on the top!?? im so confused
cscx
Your fractions aren't lining up. This question is also not calculus. Please do not post with different names.
To express the expression `1 / (csc(x) + cot(x)) + 1 / (csc(x) - cot(x))` in terms of `sin(x)`, we can start by simplifying the expression using the reciprocal identities for sine, cosecant, and cotangent.
1 / (csc(x) + cot(x)) + 1 / (csc(x) - cot(x))
First, rewrite `csc(x)` as `1 / sin(x)` and `cot(x)` as `cos(x) / sin(x):
1 / (1 / sin(x) + cos(x) / sin(x)) + 1 / (1 / sin(x) - cos(x) / sin(x))
Now, let's find a common denominator for both terms:
sin(x) / (sin(x) + cos(x)) + sin(x) / (sin(x) - cos(x))
The common denominator is `(sin(x) + cos(x))(sin(x) - cos(x))` which simplifies to `sin^2(x) - cos^2(x)`:
(sin(x)(sin(x) - cos(x))) / (sin(x) + cos(x))(sin(x) - cos(x)) + (sin(x)(sin(x) + cos(x))) / (sin(x) - cos(x))(sin(x) + cos(x))
Now, combine the terms together:
(sin(x)(sin(x) - cos(x)) + sin(x)(sin(x) + cos(x))) / (sin(x) + cos(x))(sin(x) - cos(x))
Using the distributive property, you can simplify further:
(sin^2(x) - sin(x)cos(x) + sin^2(x) + sin(x)cos(x)) / (sin(x) + cos(x))(sin(x) - cos(x))
And finally, combining like terms:
(2sin^2(x) + 2sin(x)cos(x)) / (sin(x) + cos(x))(sin(x) - cos(x))
So the expression `(1 / (csc(x) + cot(x))) + (1 / (csc(x) - cot(x)))` expressed in terms of `sin(x)` is `(2sin^2(x) + 2sin(x)cos(x)) / (sin(x) + cos(x))(sin(x) - cos(x))`.
As for expressing `(1 + cot(x)) / (1 - sin^2(x))` in terms of `cos(x)`, we'll use the reciprocal and Pythagorean identities.
First, we can simplify the expression `(1 + cot(x)) / (1 - sin^2(x))` by replacing `cot(x)` with `cos(x) / sin(x)`:
(1 + cos(x) / sin(x)) / (1 - sin^2(x))
Next, to find a common denominator, we multiply the numerator and denominator by `sin(x)`:
(sin(x) + cos(x)) / (sin(x) - sin^3(x))
Now, we can use the Pythagorean identity `sin^2(x) = 1 - cos^2(x)` to express the denominator in terms of `cos(x)`:
(sin(x) + cos(x)) / (sin(x) - (1 - cos^2(x)))
Simplifying further:
(sin(x) + cos(x)) / (-sin(x) + cos^2(x))
So the expression `(1 + cot(x)) / (1 - sin^2(x))` expressed in terms of `cos(x)` is `(sin(x) + cos(x)) / (-sin(x) + cos^2(x))`.
Regarding your question about `csc(x) = 1/sin(x)`, if you note that there is an extra `1` on the numerator of the first fraction, you can bring it to the denominator by multiplying the `1` by `sin(x)`. This will result in the numerator becoming `sin(x)(1) = sin(x)`.